凸分析及优化理论在复杂摩擦市场的无套利分析中的应用

项目名称： 凸分析及优化理论在复杂摩擦市场的无套利分析中的应用

项目编号： No.70801068

项目类型： 青年科学基金项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 董洪斌

作者单位： 中央财经大学

项目金额： 18万元

中文摘要： 本项目主要对复杂摩擦金融市场的无套利条件加以研究：首先，建立适当的数学模型刻画实际金融市场中复杂的摩擦，利用一个凸规划刻画了无套利条件，并讨论了无套利条件与状态价格向量之间的关系，利用优化理论中的对偶问题刻画了摩擦市场下的无套利条件以及讨论了无套利条件与期限结构之间的关系； 其次，在新的摩擦市场下，结合向量优化理论中有效点的概念，提出新的无套利概念并刻画它，研究其与其它无套利条件之间的关系； 第三，在无套利条件下，研究了最优再保险策略，分别最小化保险公司剩余风险的VaR和CTE两个风险策略，讨论了组合比例与超额损失再保险的最优形式，并讨论了风险调整资本收益率下的最优再保险策略；第四，运用无套利定价理论，结合工程领域的HHT方法，对我国股票市场中的权证定价以及存在的摩擦做实时的调研,研究了股票的风险和定价，对得出的理论进行检验。此外，本项目还对向量优化有效解的充分必要条件及保险资金的投资有效性作了研究。

中文关键词： 凸分析; 优化理论; 摩擦市场; 无套利分析; 最优再保险

英文摘要： The project studies the absence of arbitrage problem in the complicated finical market mainly from the following aspects: First is the characterization of no-arbitrage in frictional markets. The project extends the martingale analysis of no-arbitrage pricing with transaction costs, bid-ask spreads and taxes. The project gives a characterization of no-arbitrage condition in terms of some convex programming problem, discusses the relationship between the no arbitrage condition and the state prices vector, characterizes the no arbitrage conditions in terms of the dual theorem of optimization and establishs the relationship between the no arbitrage conditions and the term structure. Secondly， the project introduces a new concept - robust asymptotic no free lunch coinciding with the notion of the proper efficient solutions in vector optimization theory, presents a necessary and sufficient condition for no robust asymptotic free lunch, proves the existence of a strong consistent price system, and discusses the relationship of the former notions of the no-arbitrage condition. Thirdly, under no-arbitrage condition, the project considers the problem of minimizing the VaR and CTE of an insurer's retained risk by controlling the combinational quota-share and stop-loss reinsurance strategy, and discusses the reinsurance strategies under return on risk-adjusted capital. Finally, the project introduces the new signal process method HHT into the option price under no-arbitrage condition. HHt method is used to simulation and computing and the volatility of price. Moreover, the project also involves the necessary and sufficient condition of the proper efficient solution in the vector set-valued optimization and the investment efficiency of the insurance fund.

英文关键词： Convex Analysis；Optimization theory；Frictional Markets；No-arbitrage；Optimal reinsurance；